aa r X i v : . [ m a t h . G M ] A ug Prime Values of Quadratic Polynomials
N. A. Carella
Abstract : This note investigates the prime values of the polynomial f ( t ) = qt + a forany fixed pair of relatively prime integers a ≥ q ≥ x ≥
1, an asymptotic result of the form P n ≤ x / , n odd Λ( qn + a ) ≫ qx / / ϕ ( q )is achieved for q ≪ (log x ) b , where b ≥ The basic problem of prime values of linear polynomials f ( t ) = qt + a ∈ Z [ t ] is completelysolved. Dirichlet theorem for primes in arithmetic progressions proves that any admissiblelinear polynomial has infinitely many prime values. The quantitative form of this theoremhas the asymptotic formula X n ≤ xn ≡ a mod q Λ( n ) ∼ ϕ ( q ) x, (1)where gcd( a, q ) = 1, as x → ∞ . The next basic problem of prime values of quadraticpolynomials f ( t ) = at + bt + c ∈ Z [ t ] has very precise heuristics and many partial results,but there is no qualitative nor quantitative results known. This note investigates the primevalues of the admissible quadratic polynomials f ( t ) = qt + a ∈ Z [ t ], and proposes thefollowing result. Theorem 1.1.
Let x ≥ be a large number. Let a and q be a pair of relatively primeintegers, with opposite parity, and q ≪ (log x ) b , where b ≥ is a constant. Then, X n ≤ x / n odd Λ( qn + a ) ≫ q ϕ ( q ) x / + O (cid:16) x / e − c √ log x (cid:17) , (2) where c > is an absolute constant. The core of the proof in Section 2 consists of the quadratic to linear identity in Section6, and other results proved in Section 3 to Section 8. Theorem 1.1 proves the predictedasymptotic formula X n ≤ x / n odd Λ( qn + a ) ∼ c f x / , (3)but not the constant c f ≥
0, see [3] for finer details. The conjectured constant, whichdepends on the polynomial f ( t ) = qt + a , has the form c f = ǫ Y p ≥ p | q (cid:18) pp − (cid:19) Y p ≥ p ∤ q (cid:18) − (cid:18) − aqp (cid:19) p (cid:19) , (4) September 2, 2020
MSC2020 : Primary 11N32, Secondary 11N05.
Keywords : Distribution of Primes; Polynomial Prime Values; Bateman-Horn Conjecture. ǫ = (cid:26) / q , q ≡ . (5)The conjectured general formula for the constant c f ( a, c ) ≥ f ( t ) = at + bt + c ∈ Z [ t ] appears in [16, p. 46], [3, p. 364], et alii.Discussions on the convergence of the product (4) appears in [3], [13, Section 5], et alii.Results on the average value c f ( a, c ), and other properties appear in [5], [25], et cetera,optimization and numerical techniques appear in [18], and similar references.The result in Theorem 1.1 is a special case of the Bateman-Horn Conjecture for poly-nomials over the integers, see [3], and [13] for a survey. Some references on the vastliterature on the theory of prime values of polynomials are provided here. The generalcircle methods are introduced in [22], [28], and the heuristics for admissible quadraticpolynomials was proposed in [16, p. 46]. More recent discussions are given in [26, p. 406],[23, p. 342], et cetera. Some partial results are proved in [15], [20], [5], [8], [17], [19],and the recent literature. The results for the associated least common multiple problemlog lcm[ f (1) f (2) · · · f ( n )] appears in [6], et cetera. The related problem for almost primesappears in [17], [19], et alii. Topics on the Bateman-Horn Conjecture for polynomialsover numbers fields and functions fields appear in [4], [7], [10], et alii, and for multivari-able polynomials appears in [9], et cetera. A very recent proof for certain collection ofquadratic polynomials f ( t ) = a ( u ) t + b ( u ) t + c ( u ) ∈ F q [ u ][ t ] over function fields of oddcharacteristic is proposed in [27, Theorem 1.2]. For any pair of fixed integers 1 ≤ a ≤ q such that gcd( a, q ) = 1, the polynomial f ( t ) = qt + a is irreducible, and it has fixed divisor div( f ) = gcd( f ( Z )) = 1, see [14,p. 395] for more details.For an integer n ≥
1, the vonMangoldt function Λ : N −→ R is defined byΛ( n ) = (cid:26) log p if n = p k , n = p k , (6)where n = p k is a prime power, and the Euler totient function ϕ : N −→ Q is defined by ϕ ( n ) = n Q p | n (1 − /p ). A primes counting function, weighted by Λ( n ), is defined by ψ ( x, q, a ) = X n ≤ x / n odd Λ( qn + a ) . (7) Proof. (Theorem 1.1): Given a large number x ≥
1, let p ≡ x < p , and let N = 2 p . Further, assume that [ x / ] = 2 k is an even integer. Now,in terms of the quadratic to linear identity in Lemma 6.1, the weighted primes countingfunction has the form X n ≤ x / n odd Λ( qn + a ) = 1 ϕ ( N ) X n ≤ xn odd Λ( qn + a ) X s ≤ x / , X ≤ u
Lemma 3.1.
Given a large number x ≥ , let p ≡ be a large prime such that x < p , and let N = 2 p . Further, assume that [ x / ] = 2 k is an even integer. Let a and q be a pair of relatively prime integers, with opposite parity, and q ≪ (log x ) b , where b ≥ is a constant. Then, ϕ ( N ) X n ≤ xn odd Λ( qn + a ) X ≤ u
Lemma 4.1.
Given a large number x ≥ , let p ≡ be a large prime such that x < p , and let N = 2 p . Further, assume that [ x / ] = 2 k is an even integer. Then, ϕ ( N ) X n ≤ xn odd Λ( qn + a ) X ≤ u Given a large number x ≥ , let p ≥ be a large prime such that x < p ,and let N = 2 p . Further, assume that [ x / ] = 2 k is an even integer. If n ≤ x is a fixedodd integer, then, Q ( n ) = 1 ϕ ( N ) X ≤ s ≤ x / , X ≤ u ≤ N − u,N )=1 e i π ( s − n ) u/N = (cid:26) if n = s , if n = s . (20) Proof. Assume n = s is a square . The hypothesis n ≤ x implies that the equation s − n = 0 has a unique integer solution s ∈ [1 , x / ] for each square integer n ∈ [1 , x ].Thus, the double finite sum has the value1 ϕ ( N ) X ≤ s ≤ x / , X ≤ u ≤ N − u,N )=1 e i π ( s − n ) u/N = 1 . (21) Assume n = s is not a square . The hypothesis N = 2 p , with p > n ≤ x and s ≤ x / , imply that the Ramanujan sum has the value c N ( s − n ) =5 − s for s − n = 0, see Lemma 7.1. Hence,1 ϕ ( N ) X ≤ s ≤ x / , X ≤ u ≤ N − u,N )=1 n = s e i π ( s − n ) u/N = 1 ϕ ( N ) X ≤ s ≤ x / n = s c N ( s − n ) (22)= 0 , the last equality follows from Lemma 7.2 since [ x / ] = 2 k is an even integer. (cid:4) This technique is very flexible, and has the advantages of being easily extended to otherclasses of integer powers as cubic integers, and quartic integers, et cetera. The quadratic to linear inequality trades off the evaluation of P n ≤ x / , odd n Λ( qn + a ) forthe evaluation of a product of some exponential sums and P n ≤ x, odd n Λ( qn + a ). Lemma 6.1. Given a large number x ≥ , let p ≥ be a large prime such that x < p ,and let N = 2 p . Further, assume that [ x / ] = 2 k is an even integer. If a and q is a pairof relatively prime integers, and opposite parity, then, X n ≤ x / odd n Λ( qn + a ) = 1 ϕ ( N ) X n ≤ xn odd Λ( qn + a ) X s ≤ x / , X ≤ u Summing the product of Λ( qn + a ) and the characteristic function Q ( n ) of squareodd integers n ≤ x return X n ≤ xn odd Λ( qn + a ) Q ( n ) = X n ≤ xn odd Λ( qn + a ) 1 ϕ ( N ) X s ≤ x / , X ≤ u Lemma 7.1. Given a large number x ≥ , let p ≥ be a large prime such that x < p ,and let N = 2 p . Further, assume that [ x / ] = 2 k is an even integer. If n ≤ x is a fixedodd integer, and s ≤ x / , then,1. c N ( s − n ) = X ≤ u Given a large number x ≥ , let p ≥ be a large prime such that x < p ,and let N = 2 p . Further, assume that [ x / ] = 2 k is an even integer. If n ≤ x is an oddinteger, then,1. X s ≤ x / s − n =0 c N ( s − n ) = (cid:26) if [ x / ] = 2 k, − if [ x / ] = 2 k ± . X s ≤ x / s − n =0 c N ( s − n ) = (cid:26) if [ x / ] = 2 k, − if [ x / ] = 2 k ± . Proof. (1) For a fixed odd integer n = s ≤ x , and s ≤ x / such that s − n = 0, the finitesum c N ( s − n ) = ( − s , see Lemma 7.1. Thus, X s ≤ x / s − n =0 c N ( s − n ) = X s ≤ x / ( − s = (cid:26) x / ] = 2 k, − x / ] = 2 k ± . (27)(2) The same proof applies to this case. (cid:4) Theorem 8.1. (Gauss) If N ≥ is an integer, then, X ≤ s ≤ N − e i πs /N = √ N if N ≡ , if N ≡ ,i √ N if N ≡ , (1 + i ) √ N if N ≡ . (28) Proof. A proof based on finite Fourier transform appears in [2, Theorem I.1.1], and a proofbased on the Poisson summation formula appears in [21, Corollary 9.16]. (cid:4) Theorem 8.2. If N ≥ is an integer, and ( n | N ) is the quadratic symbol modulo N ,then, X ≤ s ≤ N − s,N )=1 (cid:16) sN (cid:17) e i πs/N = √ N if N ≡ , if N ≡ ,i √ N if N ≡ , (1 + i ) √ N if N ≡ . (29)7 roof. Use the quadratic symbol (9) to remove the nonlinear term from the exponentialsum: X ≤ s Let x ≥ be a large number. Then X n ≤ x / Λ( n + 1) ≫ x / O (cid:16) x / e − c √ (log x ) b (cid:17) , (32) where c > is an absolute constant.Proof. Consider the polynomial f ( t ) = 4 t + 1 ∈ Z [ t ], where q = 4 and a = 1. Then, X n ≤ x / Λ( n + 1) = X n ≤ x / / Λ(4 n + 1) + O (log x ) (33) ≥ X n ≤ x / / n odd Λ(4 n + 1) ≫ ϕ (4) x / O (cid:16) x / e − c √ (log x ) b (cid:17) , where c > (cid:4) The standard heuristic for the prime values of the polynomial f ( t ) = t + 1 ∈ Z [ t ] predictsthe followings data. Conjecture 9.1. ([16]) Let x ≥ be a large number. Let Λ be the vonMangoldt function,and let χ ( n ) = ( n | p ) be the quadratic symbol modulo p . Then X n ≤ x / Λ (cid:0) n + 1 (cid:1) = c f x / + O x / log x ! , (34)8 here the density constant c f = Y p ≥ (cid:18) − χ ( − p − (cid:19) = 1 . . . . . (35)A list of the prime values of the polynomial f ( t ) = t + 1 is archived in OEIS A002496. References [1] Apostol, Tom M. Introduction to analytic number theory . Undergraduate Texts inMathematics. 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